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I have a few questions on the topic of extrema and derivatives:

  1. Consider a differentiable function $f:\mathbb{R}\to\mathbb{R}$ that reaches its maximum in $a\in\mathbb{R}$. This implies that $f'(a)=0$.

I think this is true. We now that $f$ is differentiable over $\mathbb{R}$, so $f'(a)$ exists. Because a (global) maximum is always a local maximum, we can conclude that $f'(a)=0$.

  1. If $f \in C^1([0,1])$ reaches its maximum in a point $a\in [0,1]$, then $f'(a)=0$.

This isn't necessarily true when $a=0$ or $a=1$, because then we're not working with an interior point ($f$ is differentiable over $]0,1[$).

Are my answers and way of thinking correct? Thanks for helping!

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